We probe the top-quark electric dipole moment (EDM) in top-quark pair
production via photon-photon fusion at a photon linear collider. We show how
linearly-polarized photon beams can be used to extract information on the
top-quark EDM without the use of complicated angular correlations of top-quark
decay products. If the luminosity of the laser back-scattered photon-photon
collisions is comparable to that of the $e^+e^-$ collisions, then the
measurement of the top-quark EDM obtained by counting top-quark-production
events in photon fusion can be as accurate as the measurement obtained by
studying the $t\bar{t}$ decay correlations in $e^+e^-$ collisions using a
perfect detector.Comment: Latex, 11 pages, 1 figure (not included). One compressed postscript
  file of the paper available at
  ftp://ftp.kek.jp/kek/preprints/TH/TH-443/kekth443.ps.g

Anlauf

Atwood

Bernreuther

Ginzburg

Grazadkowski

K. Hagiwara

Kane

Krämer

Kühn

Mahmood

S.Y. Choi

Schmidt

Soni

Physics Letters B

English

arXiv

We probe the top-quark electric dipole moment (EDM) in top-quark pair production via photon-photon fusion at a photon linear collider. We show how linearly-polarized photon beams can be used to extract information on the top-quark EDM without the use of complicated angular correlations of top-quark decay products. If the luminosity of the laser back-scattered photon-photon collisions is comparable to that of the e^+e^- collisions, then the measurement of the top-quark EDM obtained by counting top-quark-production events in photon fusion can be as accurate as the measurement obtained by studying the t\bar{t} decay correlations in e^+e^- collisions using a perfect detector

Choi, S Y

Hagiwara, K

CERN Document Server

H
EP
-P
H
-9
50
64
30
KEK-TH-443
KEK Preprint 95-80
June 27, 1995
H
Probing the Top-Quark Electric Dipole Moment
at a Photon Linear Collider
S.Y. Choi and K. Hagiwara
Theory Group, KEK, Tsukuba, Ibaraki 305, Japan
Abstract
We probe the top-quark electric dipole moment (EDM) in top-quark pair pro-
duction via photon-photon fusion at a photon linear collider. We show how
linearly-polarized photon beams can be used to extract information on the top-
quark EDM without the use of complicated angular correlations of top-quark
decay products. If the luminosity of the laser back-scattered photon-photon
collisions is comparable to that of the e
+
e
 
collisions, then the measurement
of the top-quark EDM obtained by counting top-quark-production events in
photon fusion can be as accurate as the measurement obtained by studying
the t

t decay correlations in e
+
e
 
collisions using a perfect detector.
1
A detailed study of top-quark physics is one of the most important tasks of a
linear e
+
e
 
collider. The fact that the top quark has a much larger mass than the other
quarks and leptons suggests that it may be susceptible to eects from physics beyond the
Standard Model (SM), to which lighter quarks and leptons are not sensitive.
Detailed studies have been performed mainly in the processes e
+
e
 
! t

t[1, 2]
including general studies of tt, ttZ and tbW couplings[3, 4, 5, 6]. At a photon linear
collider we can measure the tt and tbW couplings. In this paper we investigate the
possibility of extracting the eective couplings of the top quark to the photon by employing
linearly polarized photons generated by the Compton back-scattering of linearly polarized
laser light on electron/positron beams of linear e
+
e
 
(or e
 
e
 
) colliders.
One may employ two methods to extract the top-quark eective couplings at a
photon linear collider. One method makes use of the quasi-freely decaying property of
the top quark by measuring various spin correlations in the t

t nal system, (bW
+
)(

bW
 
)
or (bf
1

f
2
)(

bf
3

f
4
). The other method is to employ linearly-polarized photon beams to
measure various polarization asymmetries of the initial states. It is, of course, possible to
combine the two methods. The former technique is essentially the same as that employed
in e
+
e
 
collisions[3, 4, 5, 6] with one dierence; in e
+
e
 
collisions the spin of the t

t system
is restricted to J = 1, while in photon fusion J = 0 or J  2 is allowed. Additionally it is
easy to produce linearly polarized photon beams through the Compton back-scattering of
polarized laser light o the initial electron/positron beams. Hence a  collider provides
some unique opportunities. We will show that CP-odd correlations can be measured by
a counting experiment where no information on the momenta and polarization of the top
quark decay products are needed.
A CP-odd asymmetry can be formed at a -collider when the following J = 0
amplitudes of two photons in the CP-even and CP-odd states are both non-vanishing;
M [ ! X(CP = +)] / ~
1
 ~
2
; M [ ! X(CP =  )] / (~
1
 ~
2
) 
~
k
1
; (1)
where 
1
and 
2
are the polarizations of the two colliding photons and
~
k
1
is the momentum
vector of one photon in the  c.m. frame. This fact has been used in Ref. [7] to show
that a Higgs boson (X = H) of mixed CP character could be distinguished from a pure
2
CP eigenstate by the fact that a certain asymmetry between production cross sections
initiated by colliding photons of dierent helicities is non-zero. A similar analysis can be
applied to the t

t production process (X = t

t) where, for instance, the top-quark EDM
generates CP-odd terms. It is evident that the determination of the CP of  system
requires linearly polarized colliding photons.
In general, a linearly polarized photon state is a superposition of two helicity states
with equal weight. The state vector of a linearly polarized photon with the polarization
along an azimuthal angle  in a given coordinate system can be written as
ji =
1
p
2
h
 e
 i
j = +i+ e
i
j =  i
i
: (2)
The state vector of the two-photon system, where two photons collide head-on, is the
direct product
j
1
; 
2
i =
1
2
h
e
 i(
1
 
2
)
j++i   e
 i(
1
+
2
)
j+ i   e
i(
1
+
2
)
j  +i+ e
i(
1
 
2
)
j    i
i
: (3)
Then the scattering amplitude from the linearly polarized state to a nal state X is
hXjM j
1
; 
2
i =
1
2
h
e
 i(
1
 
2
)
M
++
  e
 i(
1
+
2
)
M
+ 
  e
i(
1
+
2
)
M
 +
+ e
i(
1
 
2
)
M
  
i
: (4)
Here we have used for helicity amplitudes the notation
M

1

2
= hXjM j
1

2
i: (5)
The azimuthal angle dierence,  = 
1
 
2
, is not changed with respect to rotation along
the two-photon direction, while the azimuthal angle sum, ' = 
1
+ 
2
, can be cancelled
by an overall rotation along the z-axis. After taking an average over the azimuthal angle
' we obtain the matrix element squared as
() =
1
4
X

1
;
2
=
jM

1

2
j
2
 
1
2
cos(2)Re
h
M
++
M

  
i
 
1
2
sin(2)Im
h
M
++
M

  
i
; (6)
which diers from the unpolarized cross section by the last two terms. Only the last term
is CP-odd.
3
We now turn to the specic process  ! t

t with Compton back-scattered laser
lights as a photon source at a future linear e
+
e
 
(e
 
e
 
) collider. The eective Lagrangian
for the top-quark{photon interaction is generalized by including terms of dimension ve:
L
M
= eQ
t
 
d
t
2m
t
!

t

F

t; L
E
=  ieQ
t
 
~
d
t
2m
t
!

t


5
F

t; (7)
where 

=
i
2
[

; 

], F

is the photon eld strength and Q
t
= 2=3 is the top-quark elec-
tric charge in units of the proton charge e. The coecient 
t
= ed
t
Q
t
=2m
t
is then called
the anomalous magnetic dipole moment (MDM) of the top quark, which receives one-loop
contributions of the order 
s
=m
t
. This would very likely mask any new-physics contri-
bution to the MDM. On the other hand, the top-quark EDM, ~
t
= e
~
dQ
t
=2m
t
, violates
T-invariance. Its presence can hence be tested unambiguously via CP-odd correlations.
Therefore, we focus in this letter on the measurement of
~
d
t
.
The top-quark EDM modies the SM t

t vertex as
 

(k; p; p
0
) =  ieQ
t
"


+
~
d
t



5
k

2m
t
#
; (8)
where k = p  p
0
is the four-momentum of the photon and p and p
0
are four-momenta of
the incoming and outgoing top quarks, respectively. With this modication, the helicity
amplitudes for the process  ! t

t are in the  c.m. frame given by
M

1

2
;
=
4Q
2
t
N
c
(1 
^

2
cos
2
)
A

1

2
;
; (9)
where
A
;
=  
p
s^
2m
t
(+ 
^
)
"
4m
2
t
s^
+
~
d
t
 
i 
~
d
t
2
!
(1 + 
^
 cos
2
)
#
;
A
;; 
=
~
d
t
 
i 
~
d
t
2
!
^
 cos  sin ;
A
; ;
=
p
s^
2m
t

^
 sin
2

"
4m
2
t
s^
  i
~
d
t

^
 +
~
d
2
t
2
#
;
A
; ;; 
=
^
 ( + cos ) sin 
"
1 
s^
8m
2
t
~
d
2
t
(1  
^

2
cos )
#
; (10)
where ;

 and =2; =2 are the two-photon and t;

t helicities, respectively. s^ is the 
c.m. energy squared,
^
 =
q
1  4m
2
t
=s^, and  is the polar angle between the photon beam
4
and the top-quark directions. The dierential cross section of the process  ! t

t is then
given by
d
d cos 
(
1

2
;
0
1

0
2
) =
4
2
s^
Q
4
t
N
c

(1 
^

2
cos
2
)
2
T

1

2
;
0
1

0
2
; (11)
T

1

2
;
0
1

0
2
=
X

1

2
=
A

1

2
;
1

2
A


0
1

0
2
;
1

2
: (12)
We nd to rst order in
~
d
t
,
1
4
X

1

2
=
T

1

2
;
1

2
=
1
2
h
1 + 2
^

2
  (1 + sin
4
)
^

4
i
;
Re[T
++;  
] =  2(1 
^

2
)
2
; Im[T
++;  
] =  4(1 
^

2
)Re(
~
d
t
): (13)
One can see from these expressions that, to rst order, the electric-dipole-form-factor,
~
d
t
,
does not contribute to the unpolarized cross-section, nor to the CP-even term, Re[T
++;  
],
but it can contribute to the CP-odd term, Im[T
++;  
]. It is now clear that its measurement
requires linear polarization of the two photon beams, which can be achieved eciently
via the Compton backscattering of laser beams[8].
The event rate for t

t production can now be obtained by folding a photon lumi-
nosity spectral function, dL

, with the t

t production cross section as
dN = dL

3
X
i;j=0
h
i


j
i


ij
; (14)
where 
i
(


i
) are the Stokes parameters (with 
0
=


0
= 1) for 
1
(
2
), and 
ij
are the
corresponding t

t production cross-sections. The linear-polarization transfer from the laser
photons to the high-energy photons is described by the 
1
and 
3
components of the Stokes
vector,
~
 = (
1
; 
2
; 
3
). The length, l =
q

2
1
+ 
2
3
, of these two components depends on
the nal state photon energy and on the value of the parameter x = 2p
e
p

=m
2
e
. dL

and
h
i


j
i

are obtained as functions of the fraction (= s^=s) of the  c.m. energy-squared, s^,
to the e
+
e
 
c.m. energy squared s by averaging over collision, including a convolution over
the energy spectra for the colliding photons. Operation below the threshold for e

pair
production in collisions between the laser beam and the Compton-backscattered photon
beam requires x  2(1+
p
2); the lower bound on x depends on the lowest available laser
5
frequency and the production threshold of the nal state. The linear-polarization transfer
is large for small values of x if the photon energy to electron beam energy, y, is close to
the maximum value, i.e.
y ! y
m
=
x
1 + x
; (15)
for a given parameter x. The photon energy spectrum is given by

0
(y) =
1
1  y
+ 1  y   4r(1  r); (16)
and the Stokes linear component, l(y) = 
3
(y)=
0
(y), follows from

3
(y) = 2r
2
; (17)
with r = y=x(1  y). The maximum value of l(y) is reached at y = y
m
,
l(y
m
) =
2(1 + x)
1 + (1 + x)
2
; (18)
and approaches unity for small values of x. Since only part of the laser polarization is
transferred to the high-energy photon beam, it is useful, for the sake of discussion, to
dene the polarization asymmetry A as
A() =
h
3

3
i

h
0

o
i

; (19)
where
h
i

i
i

=
1
N
2
(x)
Z
y
m
=y
m
dy
y

i
(y)
i
(=y): (20)
The normalization factor N (x) is dened as
N (x) =
Z
y
m
0

0
(y)dy = ln(1 + x)

1 
4
x
 
8
x
2

+
1
2
+
8
x
 
1
2(1 + x)
2
: (21)
For the linearly-polarized laser beams the number of the t

t events is given by
dN = 
2
L
ee
h
0

0
i

d
h

0
  cos(2)A
A
1
  sin(2)A
A
2
Re(
~
d
t
)
i
; (22)
where for the two-photon luminosity function the relation
dL

= 
2
L
ee
h
0

0
i

d (23)
6
is used and the respective cross sections are given by

0
(s^) =
4
2
Q
4
t
N
c
^

s^
"
 2 +
^

2
+
3 
^

4
2
^

ln
 
1 +
^

1 
^

!#
;

A
1
(s^) =
4
2
Q
4
t
N
c
^

s^
(1 
^

2
)
4
"
1 +
1 
^

2
2
^

ln
 
1 +
^

1 
^

!#
;

A
2
(s^) =
4
2
Q
4
t
N
c
^

s^
1
2
"
1 +
1 
^

2
2
^

ln
 
1 +
^

1 
^

!#
: (24)
The expression of 
0
(s^) in Eq. (24) coincides with that of Ref. [9]. The e- conversion
coecient, , in Eq. (23) is assumed to be 1 in the following analysis.  is in the e
+
e
 
(e
 
e
 
) c.m. frame the angle between the directions of maximum linear polarization of the
two laser beams, which are assumed to be approaching nearly head-on towards the electron
and positron beams, respectively. And L
ee
is the integrated e
+
e
 
(e
 
e
 
) luminosity.
The last term in Eq. (22) depends on sin(2) such that the EDM contribution can
be isolated by taking the dierence of events rates at  =

4
and  =  

4
. Then one
measure of the signicance of the EDM contribution is the number of standard deviations
(N
SD
) by which jN( =

4
)   N( =  

4
)j exceeds the expected uncertainty of the
background distribution:
N
SD
= Re(
~
d
t
)
p
2N
A
2
p
N
0
; (25)
where
N
0
= L
ee
Z

m
4m
2
t
=s
h
0

0
i


0
(s); N
A
2
= L
ee
Z

m
4m
2
t
=s
h
3

3
i


A
2
(s); (26)
with 
m
= y
2
m
in Eq. (15). At the 1- level the allowed maximum value of the EDM
parameter, Re(
~
d
t
), is
jRe(
~
d
t
)j
m
=
s
1
2N
0
N
0
N
A
2
; (27)
if no asymmetry is found. The N
SD
- level upper bound is obtained simply by multiplying
jRe(
~
d
t
)j
m
by N
SD
. We shall evaluate the EDM upper bound, jRe(
~
d
t
)j
m
for a yearly
integrated luminosity of 20 fb
 1
.
7
A crucial experimental issue is the optimal means for maximizing N
A
2
=
p
N
0
. It
requires obtaining the smallest possible value of x to make the linear polarization transfer
as large as possible. However, the large top-quark mass does not allow x to be very small.
For a given c.m. energy squared, s, the allowed range for x is given by
2m
t
p
s  2m
t
 x  2(1 +
p
2): (28)
Numerically, for m
t
= 175 GeV the minimum value of x is 7=3 at
p
s = 500 GeV and
7=13 at
p
s = 1 TeV, respectively. The EDM upper bound (27) will be very sensitive
to the value of
p
s since a small x requires a large s. Fig. 1 shows the cross section 
0
and the ratio 
A
2
=
0
with respect to the  c.m. energy,
p
s^, for m
t
= 150, 160, 170,
180, and 190 GeV. The cross section is maximal near
p
s^ = 2m
t
+ 100 GeV, while the
ratio 
A
2
=
0
is maximal at the threshold. Therefore, it is important to adjust the laser
energy and electron/positron beam energies so as to obtain as many t

t events as possible
while retaining x small so as to have large transverse polarization of the back-scattered
photons. We make a numerical analysis for
p
s = 500 GeV and m
t
= 150 GeV as well as
for m
t
= 170 GeV; this facilitates direct comparison with the results of Ref. [5]. Table 1
shows the x dependence of the number of t

t events, the ratio N
A
2
=N
0
, and the EDM
upper bound. N
0
increases while N
A
2
=N
0
decreases with x. As a result, the optimal
upper bound jRe(~)j
m
for
p
s = 500 GeV is obtained near x = 2:2 for m
t
= 150 GeV as
jRe(~)j
m
=
eQ
t
jRe(
~
d)j
m
2m
t
= 5:8 10
 18
(e  cm): (29)
The 1- upper bound (29) should be compared with the corresponding bound
jRe(~)j
m
= 7:3 10
 18
(e  cm); (30)
which is obtained from the spin correlation study[5] of the cascade process, e
+
e
 
! t

t,
t ! bW
+
(

t !

bW
 
), and W
 
! l
l
(W
+
! l
+

l
)
1
for 1:3  10
4
t

t events expected at
p
s = 500 GeV. It is now clear from the bounds (29) and (30) that, with the  luminosity
comparable to the e
+
e
 
luminosity, a counting experiment of top-quark pair production
1
The l
+
l
 
channel gives the strongest bound since charged leptons analyze the t and

t spins most
eciently when the decays are as predicted by the SM.
8
events by using linearly polarized laser beams at a photon linear collider can measure the
top-quark EDM as accurately as or more accurately than a perfect detector can achieve
by studying the t

t decay correlations in e
+
e
 
collisions. It is clear that by identifying the
t or

t mometum, we can make use of J  2 amplitudes which are large at s^ m
2
t
[10].
In summary, it is possible to measure the top-quark EDM in the polarized  mode
by counting t

t pair production events in a straightforward manner. In the e
+
e
 
mode it
is possible to measure the top-quark EDM by studying t

t decay correlations. Comparison
of our results for the  mode with previous analyses in the e
+
e
 
mode shows that the
two approaches are competitive. As no information on the momenta and polarization
of the top-quark decay products is directly required, linearly-polarized laser beams with
an adjustable beam energy provide us with a very ecient way of probing the top-quark
EDM at a photon linear collider.
Acknowledgements
The authors would like to thank R. Szalapski for careful reading the manuscript. This
work was supported in part by the Japan Society for the Promotion of Science (No.
94024) and also by the Grant-in-Aid for Scientic Research from the Japanese Ministry
of Education, Science and Culture (No. 05228104).
9
References
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+
e
 
Colliders" (Saariselka, Filand, September 1991), eds. R. Orava,
P. Eerola, and M. Nordberg (World Scientic, Singapore, 1992).
[2] Proceedings of the 2nd International Workshop on \Physics and Experiments with
Linear e
+
e
 
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10
Tables
Table 1 The ratioN
A
2
=N
0
, the number of t

t eventsN
0
, and the upper EDM bound, [Re(~
t
)]
m
,
for selected values of x with
p
s = 500 GeV, m
t
= 150 GeV and L
ee
= 20 fb
 1
. The
numbers in parentheses are for m
t
= 170 GeV.
x N
0
N
A
2
=N
0
jRe(~
t
)j
m
(10
 17
e  cm)
1:8 156 (  ) 0:33 (  ) 0:76 (  )
2:0 395 (  ) 0:26 (  ) 0:60 (  )
2:2 680 (2) 0:20 (0:31) 0:58 (7:24)
2:4 983 (36) 0:16 (0:26) 0:60 (2:01)
2:6 1290 (108) 0:13 (0:21) 0:65 (1:39)
2:8 1594 (206) 0:11 (0:18) 0:71 (1:21)
3:0 1890 (320) 0:09 (0:15) 0:79 (1:16)
3:2 2178 (445) 0:08 (0:13) 0:88 (1:16)
3:4 2456 (574) 0:06 (0:11) 0:98 (1:20)
3:6 2724 (707) 0:05 (0:09) 1:09 (1:27)
Figures
Fig. 1 The cross section 
0
(s^) and the ratio 
A
2
(s^)=
0
(s^) with respect to the  c.m energy
p
s^ for m
t
= 150, 160, 170, 180, and 190 GeV.
11
200 300 400 500 600 700 800 900 1000
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.4
0.8
1.2
1.6
 s [GeV]
σ
0 
[p
b]
σ
Α
2 
/ σ
0
mt=150 GeV
mt=160 GeV
mt=170 GeV
mt=180 GeV
mt=190 GeV


Probing the top-quark electric dipole moment at a photon linear collider

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Probing the Top-Quark Electric Dipole Moment at a Photon Linear Collider

Probing the Top-Quark Electric Dipole Moment at a Photon Linear Collider

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